
For a fixed \({\mathbf {k}_{\rm F}}\) , we substitute Eq. (REF ) into Eq. (REF ) and obtain
[1]}
\(&\int {\rm d}{\mathbf {r}}^{\prime } \Delta ({\mathbf {r}},{\mathbf {r}}^{\prime }) \psi _{\downarrow }^\dagger ({\mathbf {r}}^{\prime })\rightarrow \int {\rm d}{\mathbf {r}}^{\prime } \Delta ({\mathbf {x}},{\mathbf {R}}) \psi _{{\mathbf {k}_{\rm F}}\downarrow }^\dagger ({\mathbf {r}}^{\prime }) {\rm e}^{{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {r}}^{\prime }} \cr &= {\rm e}^{{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {r}}} \int {\rm d}{\mathbf {r}}^{\prime } \Delta ({\mathbf {x}},{\mathbf {r}}-\frac{{\mathbf {x}}}{2}) \psi _{{\mathbf {k}_{\rm F}}\downarrow }^\dagger ({\mathbf {r}}-{\mathbf {x}}){\rm e}^{-{\rm i}{\mathbf {k}_{\rm F}}\cdot ({\mathbf {r}}-{\mathbf {r}}^{\prime })} \cr &\simeq {\rm e}^{{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {r}}} \int {\rm d}{\mathbf {x}}\left[ \Delta ({\mathbf {x}},{\mathbf {r}}) - \frac{{\mathbf {x}}}{2}\cdot \nabla \Delta ({\mathbf {x}},{\mathbf {r}}) \right] \cr &~~~~~~~~~~~~~~~~\times \left[ \psi _{\mathbf {k}_{\rm F}}({\mathbf {r}}) - {\mathbf {x}}\cdot \nabla \psi _{\mathbf {k}_{\rm F}}({\mathbf {r}}) \right] {\rm e}^{-{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {x}}} \cr &\simeq {\rm e}^{{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {r}}} \int {\rm d}{\mathbf {x}}\Delta ({\mathbf {x}},{\mathbf {r}}) \psi _{\mathbf {k}_{\rm F}}({\mathbf {r}}) {\rm e}^{-{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {x}}} \cr &= {\rm e}^{{\rm i}{\mathbf {k}_{\rm F}}\cdot {\mathbf {r}}} \Delta ({\mathbf {k}_{\rm F}},{\mathbf {r}}) \psi _{\mathbf {k}_{\rm F}}({\mathbf {r}}),\) 
